A circular coil of mean radius of `7 cm` and having `4000` turns Is rotate at the rate of `1800` revolution per minute in the earth 's magnetic field (B=0.5 gauss), the maximum e.m.f. induced in coil will be

Options

A. `1.158 V`
B. `0.58 V`
C. `0.29 V`
D. `5.8 V`

Explanation

To solve the problem of finding the maximum electromotive force (e.m.f.) induced in a circular coil rotating in a magnetic field, we can follow these steps:

### Step 1: Convert the given values to appropriate units
- **Mean radius (R)**: Given as 7 cm, we convert it to meters:
\[ R = 7 \, \text{cm} = 7 \times 10^{-2} \, \text{m} \]
- **Number of turns (N)**: Given as 4000 turns, we keep it as:
\[ N = 4000 \]
- **Magnetic field (B)**: Given as 0.5 gauss, we convert it to tesla:
\[ B = 0.5 \, \text{gauss} = 0.5 \times 10^{-4} \, \text{T} \]

### Step 2: Convert the rotation speed to radians per second
- **Revolutions per minute (rpm)**: Given as 1800 rpm, we convert it to revolutions per second:
\[ \text{Revolutions per second} = \frac{1800}{60} = 30 \, \text{revolutions/second} \]
- Now, we convert revolutions per second to radians per second (since \(1 \, \text{revolution} = 2\pi \, \text{radians}\)):
\[ \omega = 30 \times 2\pi \, \text{radians/second} = 60\pi \, \text{radians/second} \]

### Step 3: Calculate the area (A) of the circular coil
- The area \(A\) of the circular coil is given by:
\[ A = \pi R^2 = \pi (7 \times 10^{-2})^2 = \pi (49 \times 10^{-4}) = 49\pi \times 10^{-4} \, \text{m}^2 \]

### Step 4: Use the formula for maximum e.m.f. induced
The formula for the maximum e.m.f. (\(E_0\)) induced in the coil is:
\[ E_0 = N \cdot B \cdot A \cdot \omega \]
Substituting the values we have:
\[ E_0 = 4000 \cdot (0.5 \times 10^{-4}) \cdot (49\pi \times 10^{-4}) \cdot (60\pi) \]

### Step 5: Simplify and calculate the value
1. Calculate \(E_0\):
\[ E_0 = 4000 \cdot 0.5 \cdot 49 \cdot \pi^2 \cdot 10^{-4} \cdot 60 \]
2. Calculate the numerical values:
\[ E_0 = 4000 \cdot 0.5 \cdot 49 \cdot 60 \cdot \pi^2 \cdot 10^{-8} \]
\[ E_0 = 4000 \cdot 0.5 \cdot 2940 \cdot \pi^2 \cdot 10^{-8} \]
\[ E_0 = 5880000 \cdot \pi^2 \cdot 10^{-8} \]

3. Using \(\pi \approx 3.14\):
\[ E_0 \approx 5880000 \cdot (3.14)^2 \cdot 10^{-8} \approx 5880000 \cdot 9.8596 \cdot 10^{-8} \]
\[ E_0 \approx 0.58 \, \text{V} \]

### Final Answer
The maximum e.m.f. induced in the coil is approximately:
\[ \boxed{0.58 \, \text{V}} \]